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Semilinear Second Order Elliptic Oscillation

Published online by Cambridge University Press:  20 November 2018

C. A. Swanson*
Affiliation:
Department of Mathematics University of British Columbia VancouverB.C. V6t 1w5
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These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form

(1)

in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Agmon, S., Douglis, A., and Niremberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. XII (1959),623-727.Google Scholar
2. Allegretto, W., On the equivalence of two types of nonoscillation for elliptic operators, Pacific J. Math. 55 (1974),319-328.Google Scholar
3. Allegretto, W., Oscillation criteria for quasilinear equations, Canad. J. Math. 26 (1974),931-947.Google Scholar
4. Allegretto, W., Oscillation criteria for semilinear equations in general domains, Canad. Math. Bull. 19 (1976),137-144.Google Scholar
5. Allegretto, W., A Kneser theorem for higher order elliptic equations, Canad. Math. Bull. 20 (1977),1-8.Google Scholar
6. Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971),125-146.Google Scholar
7. Atkinson, F. V., On second order nonlinear oscillations, Pacific J. Math. 5 (1955),643-647.Google Scholar
8. Belohorec, S., Oscillatory solutions of certain nonlinear differential equations of second order, Mat.-Fyz. Casopis Sloven. Akad. Vied. 11 (1961),250-255.Google Scholar
9. Belohorec, S., On some properties of the equation y"(x)+f(x)yα(x) = 0, 0 < α < l, Mat.-Fyz. Časopis Sloven. Akad. Vied. 17 (1967),10-19.Google Scholar
10. Coles, W. J., A simple proof of a well-known oscillation theorem, Proc. Amer. Math. Soc. 19 (1968), 507. Google Scholar
11. Courant, R. and Hilbert, D., Methods of Mathematical Physics I, Wiley (Interscience), New York, 1953.Google Scholar
12. Courant, R. and Hilbert, D., Methods of Mathematical Physics II, Wiley (Interscience), New York-London, 1962.Google Scholar
13. Erbe, L., Oscillation criteria for second order nonlinear differential equations, Ann. Mat. Pura Appl. 94 (1972),257-268.Google Scholar
14. Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations, Daniel Davey and Co., New York, 1965.Google Scholar
15. Headley, V. B., Some oscillation properties of self adjoint elliptic equations, Proc. Amer. Math. Soc. 25 (1970),824-829.Google Scholar
16. Headley, V. B. and Swanson, C. A., Oscillation criteria for elliptic equations, Pacific J. Math. 27 (1968),501-506.Google Scholar
17. Heywood, J. G., Noussair, E. S., and Swanson, C. A., On the zeros of solutions of elliptic inequalities in bounded domains, J. Differential Equations 28 (1978),345-353.Google Scholar
18. Kitamura, Y. and Kusano, T., Oscillation of second order sublinear elliptic equations, Utilitas Math., to appear.Google Scholar
19. Kreith, K., Oscillation theorems for elliptic equations, Proc. Amer. Math. Soc. 15 (1964),341-344.Google Scholar
20. Kreith, K., Oscillation Theory, Lecture Notes in Mathematics, Vol. 324, Springer Verlag, Berlin, 1973.Google Scholar
21. Kreith, K. and Travis, C. C., Oscillation criteria for self-adjoint elliptic differential equations, Pacific J. Math. 41 (1972),743-753.Google Scholar
22. Kuks, L. M., Sturm's theorem and oscillation of solutions of strongly elliptic systems, Soviet Math. Dokl. 3 (1962),24-27.Google Scholar
23. Ladyzhenskaya, O. A. and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.Google Scholar
24. Leighton, W., On self-adjoint differential equations of second order, J. London Math. Soc. 27 (1952),37-47.Google Scholar
25. Moore, R. A., The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955),125-145.Google Scholar
26. Nehari, Z., On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960),101-123.Google Scholar
27. Noussair, E. S., Oscillation of elliptic equations in general domains, Canad. J. Math. 27 (1975), 1239-1245.Google Scholar
28. Noussair, E. S. and Swanson, C. A., Oscillation theory for semilinear Schrödinger equations and inequalities, Proc. Roy. Soc. Edinburgh, Sect. A, 75 (1975/76),67-81.Google Scholar
29. Noussair, E. S. and Swanson, C. A., Oscillation of semilinear elliptic inequalities by Riccati transformations, Canad. J. Math., to appear.Google Scholar
30. Noussair, E. S. and Swanson, C. A., Positive solutions of semilinear Schrôdinger equations in exterior domains, Indiana Univ. Math. J. (submitted).Google Scholar
31. Noussair, E. S. and Swanson, C. A., Positive solutions of quasilinear equations in exterior domains, under preparation.Google Scholar
32. Okikiolu, G. O., Aspects of the Theory of Bounded Integral Operators in Lp-spaces, Academic Press, New York, 1971.Google Scholar
33. Piepenbrink, J., Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541-550.Google Scholar
34. Piepenbrink, J., A conjecture of Glazman, J. Differential Equations 24 (1977),173-177.Google Scholar
35. Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.Google Scholar
36. Reid, W. T., Riccati Differential Equations, Mathematics in Science and Engineering, Vol. 86, Academic Press, New York-London, 1972.Google Scholar
37. Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations, Mathematics in Science and Engineering, Vol. 48, Academic Press, New York, 1968.Google Scholar
38. Swanson, C. A., Nonoscillation criteria for elliptic equations, Canad. Math. Bull. 12 (1969),275-280.Google Scholar
39. Swanson, C. A., Remarks on Picone's identity and related identities, Atti Accad. Naz. Lincei Mem. CI. Sci. Fis. Mat. Natur., Sez. VIII, 11 (1972),1-15.Google Scholar
40. Swanson, C. A., Strong oscillation of elliptic equations in general domains, Canad. Math. Bull. 16 (1973),105-110.Google Scholar
41. Swanson, C. A., Picone's identity, Rend. Mat. (2) 8 (1975),373-397.Google Scholar
42. Willett, D., On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969),175-194.Google Scholar
43. Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949),115-117.Google Scholar
44. Wong, James S. W., On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975),339-360.Google Scholar